How do you integrate #int 1/sqrt((1-(x^2)/3))dx# using trigonometric substitution?

Answer 1

#sqrt3arcsin(x/sqrt3)+C#

We'll use the substitution #x=sqrt3sin(theta)#. Thus, #dx=sqrt3cos(theta)d theta#. Substituting into the integral:
#intdx/sqrt(1-x^2/3)=int(sqrt3cos(theta)d theta)/sqrt(1-(3sin^2(theta))/3)=sqrt3int(cos(theta)d theta)/sqrt(1-sin^2(theta))#
Note that #sin^2(theta)+cos^2(theta)=1#, so we see that #cos(theta)=sqrt(1-sin^2(theta))#:
#=sqrt3int(cos(theta)d theta)/cos(theta)=sqrt3intd theta=sqrt3theta+C#
From #x=sqrt3sin(theta)# we see that #theta=arcsin(x/sqrt3)#:
#=sqrt3arcsin(x/sqrt3)+C#
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Answer 2

To integrate ( \int \frac{1}{\sqrt{1 - \frac{x^2}{3}}} , dx ) using trigonometric substitution, we can let ( x = \sqrt{3} \sin(\theta) ). Then, ( dx = \sqrt{3} \cos(\theta) , d\theta ).

Substituting these expressions into the integral gives:

[ \int \frac{1}{\sqrt{1 - \frac{x^2}{3}}} , dx = \int \frac{1}{\sqrt{1 - \sin^2(\theta)}} \cdot \sqrt{3} \cos(\theta) , d\theta ]

Using the trigonometric identity ( 1 - \sin^2(\theta) = \cos^2(\theta) ), we have:

[ \int \frac{1}{\sqrt{\cos^2(\theta)}} \cdot \sqrt{3} \cos(\theta) , d\theta ]

[ = \int \frac{1}{|\cos(\theta)|} \cdot \sqrt{3} \cos(\theta) , d\theta ]

Now, depending on the interval of integration, the integral will be different. If ( \theta ) is in the interval ( \left[0, \frac{\pi}{2}\right] ), then ( \cos(\theta) ) is positive, and the integral simplifies to:

[ \int \sqrt{3} , d\theta ]

[ = \sqrt{3}\theta + C ]

If ( \theta ) is in the interval ( \left[\frac{\pi}{2}, \pi\right] ), then ( \cos(\theta) ) is negative, so we need to take the absolute value:

[ \int \frac{1}{\sqrt{1 - \frac{x^2}{3}}} , dx = \int \frac{1}{|\cos(\theta)|} \cdot \sqrt{3} \cos(\theta) , d\theta = -\sqrt{3}\theta + C ]

Thus, the antiderivative is either ( \sqrt{3}\theta + C ) or ( -\sqrt{3}\theta + C ), depending on the interval of integration.

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Answer 3

To integrate ( \int \frac{1}{\sqrt{1 - \frac{x^2}{3}}} , dx ) using trigonometric substitution, we make the substitution ( x = \sqrt{3} \sin(\theta) ). This substitution transforms the integral into a form that can be solved using trigonometric identities. The steps are as follows:

  1. Substitute ( x = \sqrt{3} \sin(\theta) ) into the integral.
  2. Calculate ( dx ) using the derivative of ( \sin(\theta) ) with respect to ( \theta ).
  3. Substitute ( dx ) and ( x ) with the new expressions in terms of ( \theta ).
  4. Simplify the integral in terms of ( \theta ).
  5. Solve the integral with respect to ( \theta ).
  6. Finally, express the result in terms of ( x ).

By following these steps, we can integrate the given expression using trigonometric substitution.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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